1. Field of the Invention
The present invention relates to a mist ejection head, an image forming apparatus comprising a mist ejection head, and a liquid ejection apparatus comprising a mist ejection head, and more particularly, to a mist ejection head which uses a high-focus low-attenuation type of reflector for a mist ejection head, and an image forming apparatus and a liquid ejection apparatus comprising such a mist ejection head.
2. Description of the Related Art
Conventionally, image forming apparatuses are known which form desired images by atomizing liquid ink to form a cloud of ink, known as an ink mist, and selectively depositing this ink mist onto a recording medium.
For example, each of Japanese Patent Application Publication No. 62-85948 and Japanese Patent Application Publication No. 62-111757 discloses an ink mist image recording apparatus which generates a charged ink mist locally from the front tip of a fine ultrasonic vibrating needle which vibrates ultrasonically in accordance with an image signal, and performs recording by depositing the ink mist selectively on a recording medium by applying an electric field to the charged ink mist.
Furthermore, for example, each of Japanese Patent Application Publication No. 2002-59540 and Japanese Patent Application Publication No. 2002-166541 discloses a liquid ejection apparatus in which an ultrasonic wave is supplied to the ink inside a cavity for storing ink provided inside an ink tank, by means of a piezoelectric transducer (oscillator) provided on the bottom surface of cavity, the ultrasonic wave is reflected by the inner walls of the cavity, which are formed with a parabolic cross-section, the reflected wave concentrates at the focal point of the parabola, thereby raising the acoustic energy density in the ink, and ink is sprayed in the form of a mist from an ejection port formed in the vicinity of the focal point.
An ultrasonic wave of the MHz order is generally used to create a mist of liquid ink. More specifically, the method of creating a mist of a liquid ink uses cavitation atomization based on a cavitation phenomenon, or capillary atomization based on a capillary wave. Using the latter method enables the generation of a mist having more uniform particle size, and it also has good energy efficiency.
In the case of capillary atomization, a capillary wave is generated by applying a planar wave from below in the direction of the free liquid surface, and if the planar wave has a frequency and amplitude at or above a certain level, then a capillary wave starts to oscillate. Consequently, as the capillary wave grows, minute liquid droplets break away from the peaks of the wave, thereby creating a mist.
In this case, as shown in Japanese Patent Application Publication No. 2002-59540 and Japanese Patent Application Publication No. 2002-166541, for example, the inner walls of a cavity of the ink tank which reflect an ultrasonic wave are designed to form a reflector having a parabolic surface shape, which increases the energy efficiency by focusing the ultrasonic wave in the region of the wavelength level.
FIG. 8 shows a cross-sectional view of a conventional mist ejection head which uses a parabolic surface-shaped reflector of this kind.
The conventional mist ejection head 100 shown in FIG. 8 comprises an ink tank 110, a cavity section 112 for storing ink which is provided inside the ink tank 110, and an ultrasonic wave generating device 114 provided in the bottom surface of the cavity section 112. The ultrasonic wave generating device 114 comprises a diaphragm 116 and a piezoelectric element 118.
The inner wall of the cavity section 112 forms a reflector (reflective wall) 120 which reflects an ultrasonic wave generated by the ultrasonic wave generating device 114. The reflector 120 has a parabolic form, with a cross-sectional shape such as that shown on the right-hand side of the drawing. Furthermore, the upper end side of the cavity section 112 forms a straight cylinder section 122 having a straight shape, and the focal point 124 of the parabola formed by the cross-sectional shape of the reflector 120 is positioned in the center of the upper part of this straight cylinder section 122.
Moreover, a nozzle plate 126 is formed on the upper side of the cavity section 112, and a nozzle 128, which is an opening for ejecting ink, is formed at a position corresponding to the focal point 124. Furthermore, an ink supply channel 130 for supplying ink to the cavity section 112 from the sides, is provided at the bottom surface of the cavity section 112.
When ejecting an ink mist, an ultrasonic wave 132 is applied (in an approximately planar shape) in parallel with the axial direction of the parabola formed by the cross-sectional shape of the reflector 120, from the ultrasonic wave generating device 114, to the ink inside the cavity section 112. The ultrasonic wave 132 is reflected by the reflector 120. Since the reflector 120 has the cross-section of a parabolic shape, the reflected ultrasonic wave 132 is concentrated at the focal point 124 of the parabola.
Moreover, since the nozzle 128 is formed at the position of the focal point 124, then the ultrasonic wave 132 is concentrated at the nozzle 128, the acoustic energy of the ink is raised at the nozzle 128, and an ink droplet is ejected in the form of an ink mist, from the nozzle 128.
In this way, in the mist ejection head 100 according to the related art, the cross-sectional shape of the reflector (reflective wall) 120 is formed to have a parabolic shape, and furthermore, as shown on the right-hand side of FIG. 8, the portion P1 which is situated on the farther side than the focal point F of the parabola with respect to the apex C (in the case of FIG. 8, the portion of the parabola below the focal point F) is used as the reflective surface.
Next, the effective amplitude focusing factor (magnification) in a mist ejection head using the conventional parabolic surface-shaped reflector shown in FIG. 8 is described below.
Here, a case is considered in which the vibrational energy of an acoustic source (the ultrasonic wave generating device 114) is focused at the inlet of the nozzle (the side of the nozzle 128 adjacent to the cavity section 112), by means of the parabolic surface-shaped reflector (reflective wall 120) shown in FIG. 8.
Firstly, the geometrical focusing factor “m” of the energy when using a parabolic surface-shaped reflector is given by the following equation, (1).m=(D2−d2)/λ2  (1)
As shown in FIG. 8, D (m) is the diameter (inlet diameter) of the inlet side It of the reflector (namely, the diameter of the bottom surface side of the cavity section 112), and d (m) is the diameter (outlet diameter) of the outlet side Ot of the reflector (namely, the diameter of the upper end side of the cavity section 112). Furthermore, λ is the wavelength of the longitudinal wave in a fluid, which is expressed by the ratio between the speed of sound in a fluid (speed of propagation of longitudinal wave) v (m/s) and the frequency f (Hz) of the sound source, as indicated by the following equation, (2).λ=v/f  (2)
Moreover, the vibrational energy of a continuous body is directly proportional to the square of the amplitude, and the amplitude amplification rate is the square root of m, or √(m).
Therefore, the effective amplitude focusing factor Γ is defined by the following equation, (3), as the product of the geometric focusing factor of the amplitude due to the reflector, √(m), and the transmissibility T based on the viscous damping.Γ=T×√(m)  (3)
Here, the transmissibility T is given by the following equation, (4).T=exp(−αL)  (4)
Here, the coefficient α is α=0.8361×μf2×10−13 (neper·m−1), L(m) is the propagation distance, and μ(cP) is the coefficient of viscosity.
By consolidating the aforementioned equations, the effective amplitude focusing factor Γ is given by the following equation, (5).Γ=f√(D2−d2)/v×exp(0.8361×μf2×10−13×L)  (5)
Next, the shape of the reflector (reflective wall 120) is described below. As shown in FIG. 9, the reflector (reflective wall 120) has a parabolic surface shape, having a cross-sectional shape which comprises a portion of a parabola having axial symmetrical (as shown on the left-hand side of FIG. 9) on the far side of the focal point F of the parabola from the apex C of same (in the case of the parabola shown in FIG. 9, the portion below the focal point F).
Here, in order to describe the reflector (reflective wall) 120 in terms of an equation, the parabolic surface shape of the reflector is expressed by the following equation, (6), using the coordinate axes r and z, as shown on the right-hand side in FIG. 9.z(r)=(g+p+u+q)−(a1/RB1)r2  (6)
The meaning of the respective symbols used in the equation (6) is as described below.
Firstly, as shown in FIG. 9, the radius (inlet radius) of the inlet side It of the reflector (the bottom surface side of the cavity section 112) is taken to be RB1, the radius (outlet radius) of the outlet side Ot of the reflector (the upper side of the cavity section 112) is taken to be RA1, the height of the parabola is taken to be h as illustrated on the left-hand side of FIG. 9, and the distance between the focal point F and the apex C is taken to be p.
In this case, a1 is defined by the following equation, (7).h=a1RB1  (7)
If a1 is defined in this way, then by means of a simple calculation, the coefficient of the quadratic term of this parabola (the coefficient of r2), A, is a1/RB1.
Moreover, since the distance p between the focal point F of the parabola and the apex C of same is generally expressed by 1/(4A), using the coefficient A of the quadratic term (r2), then this distance p can be expressed as shown below in the equation (8), using the above results.p=RB1/(4a1)  (8)
Furthermore, the length q of the reflector in the axial direction shown on the right-hand side of FIG. 9 is expressed by the following equation, (9).q=(a1/RB1)×(RB12−RA12)  (9)
Moreover, from the drawing, the length g of the straight cylinder portion 122 of the cavity section 112 in the axial direction is expressed by the following equation, (10).g=h−(p+q)  (10)
Furthermore, as shown in FIG. 9, it is known that if the reflector has the shape of a parabolic surface, then the propagation distance L1 of the ultrasonic wave until reaching the focal point F, after the ultrasonic wave has entered in parallel to the axis of the parabolic surface from the inlet side It of the cavity section 112 and has been reflected by the parabolic surface, is generally a characteristic property of the parabolic surface and is a uniform distance regardless of the position of reflection. Consequently, looking in particular at a case where the ultrasonic wave is reflected at the point of radius RA1 on the outlet side Ot of the cavity section 112, from the drawing, it can be seen that the distance L1 can be expressed by the following equation, (11).L1=q+√(RA12+g2)  (11)
Here, if calculation is made by formulas (7) to (10), then the following equation, (12), is obtained.L1={(4a12+1)/4a1}×RB1  (12)
Furthermore, it can be seen that this expression can be further developed to give L1=a1RB1+RB1/4a1=h+p.
Moreover, the focal point F of the parabolic surface must be situated to the upper side of the outlet of the cavity section 112, in other words, at least inside the straight cylinder section 122 as shown in FIG. 9, and therefore the condition stated in the formula (13) below is necessary.g≧0  (13)
As described above, taking the diameter (inlet diameter) on the inlet side It of the reflector (the bottom surface side of the cavity section 112) to be D, and taking the diameter (outlet diameter) on the outlet side Ot of the reflector (the upper end side of the cavity section 112) to be d, then “D=2RB1” and “d=2RA1” are satisfied. Therefore, by using formulas (7) to (10) to rewrite formula (13), the following relationship, (14), is obtained. In other words, in order that the focal point F is positioned inside the straight cylinder section 122, it is necessary to satisfy the following relationship, (14).d≧D/2a1  (14)
Here, considering a case where g=0 as an ideal state, the diameter (outlet diameter) d at the outlet side Ot of the reflector (the upper end side of the cavity section 112) is redefined by the following equation, (15).d=min(d)=D/2a1  (15)
Here, min(d) is a symbol which expresses the minimum value of d.
In this way, the effective focusing factor Γ is expressed by the following equation, (16), as a function of the speed v of sound in the fluid, the frequency f of the acoustic source, the coefficient μ of viscosity, the diameter u of the ink supply channel (see FIG. 9), the diameter (inlet diameter) D on the inlet side It of the reflector, and the value a1 defined in formula (7) above.
                              Γ          ⁡                      (                          v              ,              f              ,              μ              ,              u              ,              D              ,                              a                1                                      )                          =                              fD            ⁢                                                            4                  ⁢                                      a                    1                    2                                                  -                1                                                          2            ⁢                          a              1                        ⁢            v            ⁢                                                  ⁢                          ⅇ                                                0.8361                  ×                  μ                  ⁢                                                                          ⁢                                      f                    2                                    ×                                      10                                          -                      13                                                        ⁢                  u                                +                                                                                                    4                        ⁢                                                  a                          1                          2                                                                    +                      1                                                              8                      ⁢                                              a                        1                                                                              ⁢                  D                                                                                        (        16        )            
Moreover, this equation, (16), is written as shown below in (17), taking γ to be γ=0.8361×10−13.
                                          Γ            ⁡                          (                              v                ,                f                ,                μ                ,                u                ,                D                ,                                  a                  1                                            )                                =                                    fD              ⁢                                                                    4                    ⁢                                          a                      1                      2                                                        -                  1                                                                    2              ⁢                              a                1                            ⁢              v              ⁢                                                          ⁢                              ⅇ                                                      γ                    ⁢                                                                                  ⁢                    μ                    ⁢                                                                                  ⁢                                          f                      2                                        ⁢                    u                                    +                                                                                                              4                          ⁢                                                      a                            1                            2                                                                          +                        1                                                                    8                        ⁢                                                  a                          1                                                                                      ⁢                    D                                                                                      ,                                  ⁢                  γ          =                      0.8361            ×                          10                              -                13                                                                        (        17        )            
In this way, it can be seen that if the effective focusing factor Γ is considered to be a function of D and a1 only, then the turning values in the D direction and the a1 direction are situated on a curve in the plane D-a1, as given by the following equations, (18) and (19).
                              D          ⁢                      |                                                            ∂                  Γ                                /                                  ∂                  D                                            =              0                                      =                              8            ⁢                          a              1                                            γ            ⁢                                                  ⁢            μ            ⁢                                                  ⁢                          f              2                        ⁢                                                  ⁢                          (                                                4                  ⁢                                      a                    1                    2                                                  +                1                            )                                                          (        18        )                                          D          ⁢                      |                                                            ∂                  Γ                                /                                  ∂                  a                                            ⁢                                                          ⁢              1                                      =                              8            ⁢                          a              1                                            γ            ⁢                                                  ⁢            μ            ⁢                                                  ⁢                          f              2                        ⁢                                                  ⁢                                          (                                                      4                    ⁢                                          a                      1                      2                                                        -                  1                                )                            2                                                          (        19        )            
For example, FIG. 10 shows the effective focusing factor Γ(D, a1) for various values of D and a1, when v=1500 (m/s), f=10 (MHz), μ=20 (cP) and u=0 (m). These contour-shaped curves indicating the values of the effective focusing factor Γ(D, a1) are known as “contour lines”.
In FIG. 10, two curves C1 and C2 which intersect these contour lines are depicted, and the curve C1 is a curve which gives the maximum value of Γ when the value of a1 given by equation (18) is uniform, and the curve C2 is a curve which gives the maximum value of Γ when the value of the D given by equation (19) is uniform.
The maximum value of the effective focusing factor Γ obtained for all of the values of (D, a1) is located at an intersection of these two curves C1 and C2. This is obtained by solving the following equation, (20).∂Γ/∂D=∂Γ/∂a1=0  (20)
By solving this equation, (20), then the values of D and a1 which give the maximum value of the effective focusing factor Γ, namely, D′ and a1′, are obtained as shown in the following equation, (21).
                                          D            ′                    =                                    3                                      γ              ⁢                                                          ⁢              μ              ⁢                                                          ⁢                              f                2                                                    ,                              a            1            ′                    =                                    3                        2                                              (        21        )            
Moreover, here, the maximum value of the effective focusing factor Γ, namely, max (Γ), is given by the following equation, (22).
                                                                        max                ⁢                                                                  ⁢                                  (                  Γ                  )                                            =                              Γ                ⁢                                                                  ⁢                                  (                                                            D                      ′                                        ,                                          a                      1                      ′                                                        )                                                                                                        =                                                3                                                  v                  ⁢                                                                          ⁢                  γ                  ⁢                                                                          ⁢                  uf                  ⁢                                                                          ⁢                                      ⅇ                                          1                      +                                              γ                        ⁢                                                                                                  ⁢                        μ                        ⁢                                                                                                  ⁢                                                  f                          2                                                ⁢                        u                                                                                                                                                    (        22        )            
In the case of FIG. 10, when these values are actually calculated, the results shown in the following equation, (23), are obtained. In other words, the maximum effective focusing factor Γ in this case is approximately 20.74.D′≈10.36 [mm], a′1≈0.866, max(Γ)≈20.74  (23)
However, in a mist ejection head which uses a conventional parabolic surface-shaped reflector such as that described above, since a portion of the parabola having axial symmetry (such as that shown on the right-hand side of FIG. 8) which is a far from the apex C with respect to the focal point F of same, (namely, the portion of the parabola below the focal point F in the case of FIG. 8) P1, is used as a reflector, then the area of the direct wave region which does not contribute to focusing is wasted and hence the spatial usage efficiency of the cavity section is poor.
Furthermore, if, on the other hand, the outlet diameter of the cavity section is reduced in order to reduce the direct wave region, then the propagation distance of the ultrasonic wave until the focal point of the parabola formed by the cross-sectional shape of the reflector becomes long, and therefore the effective focusing factor declines due to viscous damping.